arXiv subjects

Masaaki Imaizumi

Publications and source records attributed to Masaaki Imaizumi.

76 records · Page 5Linked to original sources

arXiv2017

A simple method to construct confidence bands in functional linear regression

This paper develops a simple method to construct confidence bands, centered at a principal component analysis (PCA) based estimator, for the slope function in a functional linear regression model with a scalar response variable and a functional predictor variable. The PCA-based estimator is a series estimator with estimated basis functions, and so construction of valid confidence bands for it is a non-trivial challenge. We propose a confidence band that aims at covering the slope function at "most" of points with a prespecified probability (level), and prove its asymptotic validity under suitable regularity conditions. Importantly, this is the first paper that derives confidence bands having theoretical justifications for the PCA-based estimator. We also propose a practical method to choose the cut-off level used in PCA-based estimation, and conduct numerical studies to verify the finite sample performance of the proposed confidence band. Finally, we apply our methodology to spectrometric data, and discuss extensions of our methodology to cases where additional vector-valued regressors are present.

math.ST↗

arXiv2017

PCA-based estimation for functional linear regression with functional responses

This paper studies a regression model where both predictor and response variables are random functions. We consider a functional linear model where the conditional mean of the response variable at each time point is given by a linear functional of the predictor variable. In this paper, we are interested in estimation of the integral kernel $b(s,t)$ of the conditional expectation operator, where $s$ is an output variable while $t$ is a variable that interacts with the predictor variable. This problem is an ill-posed inverse problem, and we consider two estimators based on the functional principal component analysis (PCA). We show that under suitable regularity conditions, an estimator based on the single truncation attains the convergence rate for the integrated squared error that is characterized by smoothness of the function $b (s,t)$ in $t$ together with the decay rate of the eigenvalues of the covariance operator, but the rate does not depend on smoothness of $b(s,t)$ in $s$. This rate is shown to be minimax optimal, and consequently smoothness of $b(s,t)$ in $s$ does not affect difficulty of estimating $b$. We also consider an alternative estimator based on the double truncation, and provide conditions under which the alternative estimator attains the optimal rate. We conduct simulations to verify the performance of PCA-based estimators in the finite sample. Finally, we apply our estimators to investigate the relation between the lifetime pattern of working hours and total income, and the relation between the electricity spot price and the wind power infeed.

math.ST↗

arXiv2016

Approximation method for discrete Markov decision models with a large state space

To solve discrete Markov decision models with a large number of dimensions is always difficult (and at times, impossible), because size of state space and computation cost increases exponentially with the number of dimensions. This phenomenon is called "The Curse of Dimensionality," and it prevents us from using models with many state variables. To overcome this problem, we propose a new approximation method, named statistical least square temporal difference (SLSTD) method, that can solve discrete Markov decision models with large state space. SLSTD method approximate the value function on the whole state space at once, and obtain optimal approximation weight by minimizing temporal residuals of the Bellman equation. Furthermore, a stochastic approximation method enables us to optimize the problem with low computational cost. SLSTD method can solve DMD models with large state space more easily than other existing methods, and in some cases, reduces the computation time by over 99 percent. We also show the parameter estimated by SLSTD is consistent and asymptotically normally distributed.

stat.CO↗

arXiv2016

Doubly Decomposing Nonparametric Tensor Regression

Nonparametric extension of tensor regression is proposed. Nonlinearity in a high-dimensional tensor space is broken into simple local functions by incorporating low-rank tensor decomposition. Compared to naive nonparametric approaches, our formulation considerably improves the convergence rate of estimation while maintaining consistency with the same function class under specific conditions. To estimate local functions, we develop a Bayesian estimator with the Gaussian process prior. Experimental results show its theoretical properties and high performance in terms of predicting a summary statistic of a real complex network.

stat.ML↗